Diff_UV (Differentiable Underwater Vehicle Model)

A differentiable Underwater vehicles dynamic model in 6 DOFs based on casadi operations.

This library provides capabilities to retrieve symbolic representations of kinematics and dynamics of an underwater vehicle. This makes it possible to solve for gradients, hessians and jacobians using casadi. Through casadi, the dynamics can be interfaced with state-of-the-art codes such as SUNDIALS, IPOPT, FATROP etc for integration and optimisation. It is also possible to do code generation to be used from C++, Python or Matlab/Octave.

Differentiable models are essential in underwater robotics as they enable efficient optimization, learning, and control. They allow for gradient-based optimization in control tasks, adaptation, estimation, and support real-time decision-making with Model Predictive Control (MPC). Differentiable models also enable faster simulations and more efficient analysis of the complex, nonlinear underwater dynamics.

The following assumptions have been made for the dynamics of the underwater vehicle model:

• Operates at relative low speeds (i.e. less than 2 m/s), lift forces can be neglected.
• Assumed to have port-starboard symmetry and fore-aft symmetry; and the centre of gravity (CG) is assumed to be located in the symmetry planes.
• Assumed to be hydrodynamically symmetrical about 6-DoF. Accordingly, the motions between DoFs of the vehicle in hydrodynamic can be decoupled.
• Assumed to operate below the wave-affected zone. As a result, disturbances of waves on the vehicle are negligible.

Getting Started

To use Diff_UV in your own project, simply clone this repository to your workspace:

cd path/to/src
git clone https://github.com/edxmorgan/diff_uv.git

All kinematics & hydrodynamic terms implemented in this project have been defined using Fossen's formulations. The terms implemented include:

• Kinematics : Rotation & Coordinate Transformation Matrices
• Mass: rigid body inertia and added mass in body, ned and quaternion.
• Coriolis: centripetal, coriolis, and added coriolis in body, ned and quaternion.
• Damping: linear and quadratic damping in body, ned and quaternion.
• Restoring forces: buoyancy and gravitational forces in body, ned and quaternion.
• Forward dynamics
• Inverse dynamics

Each of the aforementioned terms provide their own distinct data methods for independent use and are managed altogether within the diffUV class.

from diffUV import dyn_body,dyned_eul, kin
uv_dyn = dyn_body()
uv_dyned = dyned_eul()

inertia_mat = uv_dyn.body_inertia_matrix()
coriolis_mat = uv_dyn.body_coriolis_centripetal_matrix()
restoring_vec = uv_dyn.body_restoring_vector()
dampn_mat = uv_dyn.body_damping_matrix()

v_dot = uv_dyn.body_forward_dynamics()

For detailed usage examples of the Diff_UV, see Jupyter notebook.

Extending with CasADi Capabilities

All expressions obtained from the diffUV methods are of CasADi type. This allows them to be integrated with CasADi's advanced functionalities for optimization, symbolic computations, and numerical integrations.

Symbolic Differentiation

Utilize CasADi's automatic differentiation to compute derivatives:

from casadi import jacobian
accel_jacobian = jacobian(v_dot, uv_dyn.body_state_vector)

Code Generation

Expressions can be directly exported to MATLAB and C++ formats, for integration with external systems and applications.

import os
from casadi import Function
from diffUV.utils.symbols import *

I_o = vertcat(I_x, I_y, I_z,I_xz) # rigid body inertia wrt body origin
decoupled_added_m = vertcat(X_du, Y_dv, Z_dw, K_dp, M_dq, N_dr) # added mass in diagonals
coupled_added_m =  vertcat(X_dq, Y_dp, N_dp, M_du, K_dv) # effective added mass in non diagonals 


M_func = Function('M_b', [m, I_o, z_g, decoupled_added_m, coupled_added_m], [inertia_mat]) # for both numerical & symbolic use
M_func.generate("M_b.c")
os.system(f"gcc -fPIC -shared M_b.c -o libM_b.so")

// C++ (and CasADi)
#include <casadi/casadi.hpp>
using namespace casadi;

void diffuv_usage_cplusplus(){
  std::cout << "---" << std::endl;
  std::cout << "Usage from CasADi C++:" << std::endl;
  std::cout << std::endl;

  // Use CasADi's "external" to load the compiled function
  Function f = external("M_b", "libM_b.so");

  // Use like any other CasADi function
  double m = 11.5;
  std::vector<double> Io = {0.16, 0.16, 0.16, 0};
  double z_g = 0.02;
  std::vector<double> added_m = {-5.5 , -12.7 , -14.57,  -0.12,  -0.12,  -0.12};
  std::vector<double> coupl_added_m = {0, 0, 0, 0, 0}; // assuming decoupling motion
  std::vector<DM> arg = {m, Io, z_g, added_m, coupl_added_m};
  std::vector<DM> res = f(arg);

  std::cout << "result (0): " << res.at(0) << std::endl;
  std::cout << "result (1): " << res.at(1) << std::endl;
}

int main()
{
    diffuv_usage_cplusplus();
    return 0;
}

References

Fossen, T.I. (2011) Handbook of Marine Craft Hydrodynamics and Motion Control. John Wiley & Sons, Inc., Chichester, UK. https://doi.org/10.1002/9781119994138